Dynamically training machine-learning-based force fields for strongly anharmonic materials

Machine learning (ML) force fields have emerged as a powerful tool for computing materials properties at finite temperatures, particularly in regimes where traditional phonon-based perturbation theories fail or cannot be extended beyond the harmonic approximation. These approaches offer accuracy comparable to ab initio molecular dynamics (MD), but at a fraction of the computational cost. However, their reliability critically depends on the quality and representativeness of the training data. In particular, static training datasets often lead to failure when the force field encounters previously unseen atomic configurations during MD simulations. In this work, we present a framework for dynamically training ML force fields and demonstrate its effectiveness across materials with varying degrees of anharmonicity, including cubic boron arsenide (c-BAs), silicon (Si), and tin selenide (SnSe). Our method builds on the conventional lattice dynamics expansion of total energy and incorporates Bayesian error estimation to guide adaptive data acquisition during simulation. Specifically, we show that trajectory-averaged Bayesian errors enable efficient and targeted exploration of the configuration space, significantly enhancing the robustness and transferability of the resulting force fields. We further demonstrate how Bayesian error estimation can be applied to determine the convergence of the dynamic training without requiring additional ab initio data. This proposed framework offers a practical and easily implementable scheme to improve the training process, which is the most critical step in developing reliable ML force fields.

💡 Research Summary

The paper introduces a dynamic training framework for machine‑learning interatomic potentials (ML‑IPs) that is especially suited to strongly anharmonic materials. Traditional static training sets, which typically contain only equilibrium structures and small perturbations, often fail when the potential encounters configurations far from the training domain—a situation common in highly anharmonic systems. To overcome this limitation, the authors combine a lattice‑dynamics‑based expansion of the total energy (compressive sensing lattice dynamics, CSLD) with Bayesian error estimation derived from linear regression.

In the CSLD approach the potential energy is expressed as a series expansion in atomic displacements up to a chosen order n_max, with force‑constant tensors (FCTs) truncated by a cutoff radius r_cut. Because the expansion is linear in the FCTs, the model can be fitted by ridge regression, which naturally yields a Gaussian posterior distribution over the parameters. From this posterior the predictive distribution of forces for any configuration is a normal distribution whose variance σ² consists of two terms: a constant term (β⁻¹) reflecting the overall noise level, and a configuration‑dependent term A(u) Σ A(u)ᵀ that captures the uncertainty due to limited training data. The square root of the configuration‑dependent term is defined as the Bayesian error ε_B. Prior work has shown that ε_B correlates strongly with the true force error, allowing it to serve as an on‑the‑fly uncertainty metric without additional ab‑initio calculations.

The dynamic training loop proceeds as follows: (1) an initial dataset is generated from random structures, phonon‑mode displacements, or a short ab‑initio MD trajectory; (2) a CSLD force field is trained and used to run molecular dynamics; (3) at each MD step ε_B is evaluated; (4) if ε_B exceeds a multiple γ of the moving average μ_k